PLL: Math!

The last two posts briefly introduced what a PLL (Phase Locked Loop) does and what pieces go into making one. Now let's dive into the math. We'll start with the closed loop. 

Figure 1. Closed Loop PLL blocks.

Figure 2. Blocks with transfer functions. 

The basic transfer fuction.
φ_out / φ_in = A / (1+AB) where A is the feed-forward (K_d*F(s)*K_o/s) and B is the feedback (1/N).

Filling those variables in results in

 K_dK_oF(s)/s
1+K_dK_oF(s)/sN

There has got to be a better way way to write equations in html. If you have a suggestion, I'll try it.

Factor out the N and s to get

N K_dK_oF(s)
Ns+K_dK_oF(s)

Define a new variable K as K_dK_o/N

And consider the actual transfer function of the filter. 

Figure 3. The filter in the s-domain.

For F(s) the input is current and the output is voltage. We'll ignore the little C for now.
F(s) = (1+sCR)/sC

The transfer function is now
N sKR + K/C
s²+sKR+K/C

Control systems engineers like to write this transfer function in terms of the natural frequency and damping factor.

natural frequency = ω_n = sqrt(K/C)
damping factor = ζ = R/2*sqrt(K/C)     and if it comes up Q= 1/2ζ but I'm not going into that.

Giving a final closed loop transfer function of

φ_out = 2ζω_ns+ω_n²
φ_in      s²+ζω_ns+ω_n²

With a current starved inverter VCO as shown in the last post, the dominant intrinsic phase noise is in the VCO. The VCO noise is high pass filtered. The filter noise is bandpass filtered around ω_n. Everything else is low pass filtered. Jitter is the time domain noise; jitter in the time domain is phase noise in the frequency domain. Jitter transfer looks like figure 4. as ζ increases, the jitter peak decreases and spreads out.

Figure 4. Jitter transfer curve.

I should do the open loop stuff next time and pull this all together, maybe show some code to simplify exploring the design space. I'm more likely to drop back the trailer lights because I've been working on an improved version of them involving an actual battery charger and twice as many lights. Or my notes on buck DC-DC regulators for an interview I recently bombed.

Bruce McLaren